Not sure how familiar you are with managing OS X Server and the command line...
OS X Server does not directly implement a file drop-box mechanism as part of the package, though you can get there with what's provided.
It is possible to set up password-protected WebDAV folders, as one option using what is available in OS X Server. There's no GUI for setting that up, though. The command-line configuration steps are discussed in the 10.6 Web Services Technologies manual, and there are some support articles around, such as HT5374.
Setting up the sftp server is a different approach and is fairly easy, if you want to go that route. Give the folks a login on the server, and give them a certificate (or a password login), and they can then access their directory (either via the certificate or the password) to upload or download files. The remote clients would be using FileZilla or Cyberduck, or command-line sftp.
If you don't want to expose your server to remote attackers — the same path folks download and upload files is a likely path for remote attacks — a NAS box (preferably in a DMZ in your network) might provide you with what you want.
I haven't encountered an ftp-based dropbox implementation for OS X Server. (Full disclosure: I'm not a fan of ftp, not the least of which because of its incompatibilities with firewalls, and its use of cleartext for the user's access credentials.)
Various content-management systems likely offer this capability, either directly or via add-on modules.